The state of a free graviton is determined by its momentum \({\mathbf {p}}\)  and helicity \(\lambda \) .
The Born terms \(F_{\lambda _3\lambda _4,\lambda _1\lambda _2}\)  for the scattering of two gravitons \(|{\mathbf {p}}_1,\lambda _1\rangle |{\mathbf {p}}_2,\lambda _2\rangle \rightarrow |{\mathbf {p}}_3,\lambda _3\rangle |{\mathbf {p}}_4,\lambda _4\rangle \) ,
are taken from Ref. [1]}. The non-zero Born-term amplitudes can be found there or in Ref. [2]}.
In order to end with infrared-free PWAs we have to take into account the factor \(S_c^{-1}\)  in Eq. () up to \({\cal O}(G)\) .
This implies,


S(J)34,12=[1+i2||/44F(J)34,12]
[1-2iGsL]
+O(G2)
1+i8V(J)34,12 ,

where \(|\lambda |=|\lambda _2-\lambda _1|\) .
For instance, for \(F_{22,22}^{(0)}\)  we have that


F(0)22,22(s)=-2 s2162-1+1
dt-2
8Gs2p ,
